Log-concavity in Combinatorics

Abstract

In this thesis we survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, Alexandrov's Fenchel inequality for mixed volumes, Lorentzian polynomials, and the Hard Lefschetz theorem. We use these mechanisms to prove some new log-concavity and extremal results related to partially ordered sets and matroids. We present joint work with Ramon van Handel and Xinmeng Zeng to give a complete characterization for the extremals of the Kahn-Saks inequality. We extend Stanley's inequality for regular matroids to arbitrary matroids using the technology of Lorentzian polynomials. As a result, we provide a new proof of the weakest Mason conjecture. We also prove necessary and sufficient conditions for the Gorenstein ring associated to the basis generating polynomial of a matroid to satisfy Hodge-Riemann relations of degree one on the facets of the positive orthant.